10
10 (twelve, dozen, onety) is the base of the numeral system of this wiki. An interesting thing is 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + X + E + 10 + ... = −1/10 = −0.1, i.e. \sum_{k=1}^{\infty}(k)=-\frac{1}{10}=-0.1 (this number is \zeta(-1) , which equals the negative value of the reciprocal of 10, or the additive inverse of the multiplicative inverse of 10) 10 is a composite number, the smallest positive integer with exactly six divisors, its divisors being 1, 2, 3, 4, 6 and 10. 10 is also a highly composite number, the next one being 20 (=2×10), and all highly composite numbers ≥10 are divisible by 10. 10 is the least common multiple of the n such that anxn+an-1xn-1+...+a1x+a0 have algebraic solution: {1, 2, 3, 4}. 10 is the smallest abundant number, since it is the smallest integer for which the sum of its proper divisors (1 + 2 + 3 + 4 + 6 = 14) is greater than itself. 10 is the smallest positive integer that is not power of squarefree number. 10 is the smallest number k such that k/(largest power of squarefree kernel of k) is larger than 1, i.e. 10/6 = 2 > 1. The highly composite numbers contains the first 5 multiples of 10 (10, 20, 30, 40, and 50), and no other numbers between them. Both of 10−1 and 10+1 are primes, in fact, all of 10±1, (10/2)±1, and (10/3)±1 are primes, the next such number is 220990, besides, all such numbers must be divisible by 10. 10−2 is the smallest noncototient, 10+2 is the smallest nontotient, the average of them is 10. 10 is the first sublime number, a number that has a perfect number (6) of divisors, and the sum of its divisors is also a perfect number (24), there are only two known such numbers. 10 is the product of the first three factorials (not including 0! = 1). The square of any prime >3 is congruent to 1 mod 10 (in fact, congruent to 1 mod 20). The sum of any pair of twin primes (other than 3 and 5) is divisible by 10. The product of any pair of twin primes (other than 3 and 5) is congruent to E mod 10. 10 is the largest base such that both "all squares end with square digits" and "all primes not dividing the base end with either prime digits not dividing the base or 1" are true. 10 is the largest base such that all these three properties are true: 10 is the largest base such that both "all squares end with square digits" and "all primes end with prime digits or 1" are true. 10 is the largest natural number n'' such that for all even numbers ''k dividing n'', both ''k−1 and k''+1 are noncomposite. Regular 10-gon is the highest polygon in the tessellation. Except 0 and 1, 10 is the only natural number whose square is a Fibonacci number. (i.e. 0, 1 and 100 are the only three square Fibonacci numbers) If (and only if) there are no Wall-Sun-Sun primes, then 10 is the largest n such that the Pisano period of n2 is the same as the Pisano period of n (both of them are 20). The Frampton-Kephart primes are the primes ''p such that p''−1 (or \phi(p) , where \phi is Euler's totient function) divides 10, these primes are exactly "1 + (the even divisors of 10)" plus the only even prime (2), they are also the prime factors of 95900, the number of vertices of the 20-dimensional Leech lattice. (Note that 95900 = \prod_{p \text{ prime },\ p-1|10}(\text{ if } p|10 \text{ then } p^{6-p},\ \text{ otherwise } p) ), and the product of these primes is 16E6, which is the denominator of ''B''10 (where ''Bn'' is the ''n''th Bernoulli number), (note that ''B''10 is also the first Bernoulli number whose numerator has a prime factor ''p ≥ n''+3 (''p=497) (i.e. 10 is the smallest number n'' such that there exists prime ''p such that (p'', ''n) is an irregular pair) besides, the dozenal representation of 1/''p'' (with p'' over these primes), is terminate when ''p divides 10 (i.e. p'' = 2 or 3), and for the other three values of ''p, they are 0.2497, 0.186X35, and 0.0E, the period of them uses all the digits 0, 1, 2, ..., E exactly once. The numbers n have this property: “all primes not divide n are congruent to 1 or −1 mod n” are {1, 2, 3, 4, 6}, they are exactly the proper divisors of 10. (and the LCM of them is 10) The set of the proper divisors of 10 is the complete set of k'' in ''N such that 2\cos\frac{2\pi}{k} is in Z''. 10 is the largest natural number n such that the two sets are completely the same: {a | 0<=a A number k satisfying that “for any positive integers x,y coprime to k, x^x y (mod k) iff y^y x (mod k)” if and only if k is divisible by \lambda(k) (where \lambda(n) is the Carmichael lambda function and k is a divisor of 63X00, note that 63X00 is the largest k such that \lambda(k)=20 , and for all these k’s, \lambda(k) also divides 20, besides, the prime factors of these numbers are exactly the Frampton-Kephart primes, i.e. the primes ''p such that p''−1 (or \phi(p) , where \phi is Euler's totient function) divides 10. (equivalently, ''p−1 divides 20, since there is no prime p'' such that ''p−1 divides 20 but p''−1 does not divide 10 (both 9 and 21 are composite (9=32, 21=52, both of them are squares of primes)), in fact, the primes ''p such that p''−1 divides 10 (also 20) are exactly "1 + (the even divisors of 10)" plus the only even prime (2). 14060 is the smallest common multiple of 1, 2, 3, ..., 10, it is also the smallest number cannot be written in primorial base using only the dozenal digits (i.e. the digits 0, 1, 2, ..., E) (for factorial base, the smallest such number is 1145000000). 10 is the largest known even number expressible as the sum of two primes in only one way (5+7). 10 is the smallest (and the only known) ''n>2 such that there exist n'' consecutive integers with ''n divisors (at least start with 507711EE945E5199770213X692X2X3, there are 11 consecutive integers with 10 divisors). (The only two possible such n'' are 20 and X0) Consider the prime race mod q (where q >=2) between qn+1 and qn-1. 10 is conjectured to be the n such that the q where qn+1 first takes lead over qn-1 is largest. (such q is known for all n<=1000 except n=10 and n=20) 10 is the smallest even number ''n such that there exists a prime p'' ≥ ''n+3 (p'' = 497) such that ''p divides the numerator of B''n'' (where B''n'' is the n''th Bernoulli number). (i.e. 10 is the smallest number ''n such that there exists prime p'' such that (''p, n'') is an irregular pair) (note that the 10th prime (31) is also the smallest irregular prime) The only two known base 10 Wieferich primes are 1685 and 5E685, both of them end with 685, and it is conjectured that all base 10 Wieferich primes end with 685. The sum of this 4 fifth powers is 10X, the Xth power of 10 (i.e. 10,000,000,000): 235 + 705 + 925 + E15 = 10X, this is a counterexample of Euler's sum of powers conjecture, since 10X equals 1005, it is also a fifth power, but it only require 4 fifth powers (less than the conjectured 5) to add. (Note that 100 is the smallest number whose fifth power is the sum of 4 or less fifth powers) 10 is the smallest weight for which a cusp form exists. This cusp form is the discriminant Δ(''q) whose Fourier coefficients are given by the Ramanujan τ-function and which is (up to a constant multiplier) the 20th power of the Dedekind eta function. This fact is related to a constellation of interesting appearances of the number 10 in mathematics ranging from the value of the Riemann zeta function at −1 i.e. ζ(−1) = −1/10, the fact that the abelianization of SL(2,Z) has 10 elements, and even the properties of lattice polygons. 10 is a Pell number and a pentagonal number (5-gonal number). 10 is the number of pentominoes (5-ominoes). 1/10 = 10% is the property such that a Rubik's Cube is solvable. 10 is the least common multiple of 3 and 4, the number of sides of the first two regular polygons (equilateral triangle (3-gon) and square (4-gon)), 10 is also the least common multiple of 4 and 6, the number of faces of the first two regular polyhedrons (tetrahedron (4-hedron) and cube (6-hedron)). The centered 10-gonal numbers (centered dozagonal numbers) are exactly the star numbers. (thus, the star numbers are exactly the numbers obtained as the concatenation of a triangular number followed by a 1) A 10-sided polygon is a dozagon. A 10-faced polyhedron is a dozahedron. Regular cubes (hexahedrons, 6-faced) and octahedrons (8-faced) both have 10 edges, while regular octadozahedrons (18-faced) have 10 vertices. Two dozagons (10-gons) and one triangle can fill a plane vertex, all solutions using at least one dozagon are {3, 10, 10}, {4, 6, 10}, {3, 3, 4, 10}, and {3, 4, 3, 10}, but only the first two solutions can fill the plane. (there are 19 solutions for filling a plane vertex, but only E of them can fill the plane, the solutions are: {3, 7, 36}, {3, 8, 20}, {3, 9, 16}, {3, X, 13}, {3, 10, 10}, {4, 5, 18}, {4, 6, 10}, {4, 8, 8}, {5, 5, X}, {6, 6, 6} {3, 3, 4, 10}, {3, 4, 3, 10}, {3, 3, 6, 6}, {3, 6, 3, 6}, {3, 4, 4, 6}, {3, 4, 6, 4}, {4, 4, 4, 4} {3, 3, 3, 3, 6}, {3, 3, 3, 4, 4}, {3, 3, 4, 3, 4} {3, 3, 3, 3, 3, 3} bold for the solutions that can fill the plane, note that dozagon is the highest regular polygon in convex uniform tiling, i.e. 10 is the largest bold number in the list) 10 and 18,E27,099,E93,490,727,709,9E9,1X4,841,59E,264,E0X,X13,59X,268,567,863,258,910,E74,98X,682,454 (=(2X6)(251 − 1)(227 − 1)(217 − 1)(27 − 1)(25 − 1)(23 − 1)) are the only two known sublime numbers. (A sublime number is a positive integer which has a perfect number of positive factors (including itself), and whose positive factors add up to another perfect number, i.e. a positive integer n'' is sublime number if and only if both \sigma_0(n) and \sigma_1(n) are perfect numbers (where \sigma_k(n) is the sum of the ''k''th powers of the divisors of ''n, i.e. \sigma_k(n)=\sum_{d\mid n} d^k ), e.g. for n''=10 we have \sigma_0(10)=1^0+2^0+3^0+4^0+6^0+10^0=6 and \sigma_1(10)=1^1+2^1+3^1+4^1+6^1+10^1=24 , and both 6 and 24 are perfect numbers, thus 10 is a sublime number) Although 6 is a divisor of 10, there exists a group of order 10 (A4) without a subgroup with order 6, it is the smallest such example (i.e. 10 is the smallest number n such that there exists k dividing n and a group of order n such that this group has no subgroup with order k) All orders of non-solvable groups (thus all orders of non-cyclic simple groups) are divisible by either 10 or 18, and all orders of non-solvable groups ≤ 14000 are divisible by 10, (the smallest order of non-solvable groups not divisible by 10 is 14X28) the first two orders of non-cyclic simple groups are 50 and 120, and the greatest common divisor of them is indeed 10. All odd perfect numbers (if exist) end with 1, 09, 39, 69, or 99, and if an odd perfect number ends with 1 (i.e. = 1 mod 10), then it has at least 10 distinct prime factors. 10 is the kissing number in three dimensions (for a long time, people did not know that whether the answer is 11, this was the subject of a famous disagreement between mathematicians Isaac Newton and David Gregory). (In two demensions, it is 6 = 10÷2, and in four dimensions, it is 20 = 10×2. The values of the kissing numbers are only known in 1, 2, 3, 4, 8, and 20 dimensions (note that all such numbers of dimensions are divisors of 20), and the kissing numbers in these numbers of dimensions are 2, 6, 10, 20, 180, and 95900, however, there are also the upper bounds and the lower bounds for all other number of dimensions ≤ 20, see the list below) (note that k(1)=2 divides k(2)=3, k(2)=3 divides k(3)=10, k(3)=10 divides k(4)=20, and the three quotients are all primes, however, k(4)=20 does not divide k(5), since 34≤k(5)≤38) 10 is the number of Latin squares of order 3. 10 is the smallest number such that it is unknown whether there are ''n−1 mutually orthogonal Latin squares of order n''. (Note that there cannot be ''n mutually orthogonal Latin squares of order n'', if ''n > 1) 10 is the smallest possible order of magic square with 1 and consecutive primes starting with 3. (the smallest possible order of magic square with consecutive primes starting with 3 is 2E) In music, 10 is the number of pitch classes in an octave (an octave has 10 semitones), not counting the duplicated (octave) pitch. Also, the total number of major keys, (not counting enharmonic equivalents) and the total number of minor keys (also not counting equivalents). This applies only to 10 tone equal temperament (10-TET), the most common tuning used today in western influenced music. 10 is used for timekeeping, e.g. one year has 10 months, one day has 20 hours, one hour has 50 minutes, and one minute has 50 seconds (all of these numbers are multiples of 10). Also, there are 10 signs of the zodiac (astrological signs), and the Chinese use a 10-year cycle for time-reckoning called Earthly Branches (Chinese zodiac). 10 is the number of numbers on a clock. 10 is the number of yellow stars on the Flag of Europe. 10 is the largest possible points for rolling two dices. The series 1 + 2 + 3 + 4 + ... is divergent, however, we can use the Riemann zeta function to define its sum: −1/10 (= −0.1), since zeta(−1) = −0.1 (thus, 10 + 20 + 30 + 40 + ... (the sum of all positive multiples of 10) = −1). 0 = Aries, 1 = Taurus, 2 = Gemini, 3 = Cancer, 4 = Leo, 5 = Virgo, 6 = Libra, 7 = Scorpio, 8 = Sagittarius, 9 = Capricorn, X = Aquarius, E = Pisces. 0 = C'', 1 = ''C#/''Db'', 2 = D'', 3 = ''D#/''Eb'', 4 = E'', 5 = ''F, 6 = F#/''Gb'', 7 = G'', 8 = ''G#/''Ab'', 9 = A'', X = ''A#/''Bb'', E = B. 0 = unison, 1 = minor second, 2 = major second, 3 = minor third, 4 = major third, 5 = perfect fourth, 6 = tritone, 7 = perfect fifth, 8 = minor sixth, 9 = major sixth, X = minor seventh, E = major seventh, 10 = octave. (they are the elements in the cyclic group ''Z''10) Category:Pages